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Reconstruction of quintessence field for the THDE with

swampland correspondence in f(R, T ) gravity

Umesh Kumar Sharma

Department of Mathematics, Institute of Applied Sciences & Humanities, GLA University,Mathura-281 406, Uttar Pradesh, India

E-mail: sharma.umesh@gla.ac.in

Abstract

In the present work, we construct the Tsallis holographic quintessence model of darkenergy in f(R,T ) gravity with Hubble horizon as IR cut-off. In a flat FRW background,the correspondence among the energy density of the quintessence model with the Tsallisholographic density permits the reconstruction of the dynamics and the potentials forthe quintessence field. The suggested Hubble horizon infrared cut-off for the THDEdensity acts for two specific cases: (i) THDE 1 and (ii) THDE 2. We have reconstructedthe Tsallis holographic quintessence model in the region ωΛ > −1 for the EoS parameterfor both the cases. In addition, the quintessence phase of the THDE models is ana-lyzed with swampland conjecture to describe the accelerated expansion of the Universe.

Keywords: THDE; Quintessence; Swampland; f(R, T ) gravity.

1 Introduction

Gravity can thoroughly transform the dynamics of the physical system of the Universe, andthat is described by invariance under spatial rotations and translations, at quite big scales.The appearance of the Universe is interesting because the expansion pace of the Universehas accelerated two times, first at extremely high energies and primordial times and thesecond succeeding period is enormously still ongoing (“Dark Energy”, DE). Whenever wetry to discuss these wonders, it seems essential to reflect about the behavior of gravity itself.On referring the Universe as a whole in the elementary geometrical representation derivedfrom classical GR (General Relativity), the need for summoning might indeed cast difficultiesbecause the expansion of acceleration in Universe is two times.

ΛCDM , the standard model of cosmology depends on GR and believes that all the com-ponents of DE, which are in the form of ordinary matter, cold dark matter (CDM) and acosmological constant (Λ) created the Universe. Astonishingly, a representation of the Uni-verse is provided through ΛCDM and for this, all the data is compatible [1, 2], the modelexposes some lapses because of the problem of CC (cosmological constant). The obstaclestarts from the mismatch among Λ’s value of the observed one, from the coincidence problem

1

of late time and the one ordinarily expected from the corrections of quantum gravity [3, 4].Moreover, the model arises some sort of observational tensions between various datasets ap-pear in the ΛCDM model, for example, the linear power spectrum’s amplitude at presenttimes and the range of 8h−1Mpc, indicated through σ8,0 and on the values of the Hubble con-stant H0(= 100hkms−1Mpc−1). Comparing some examples as confined values of H0 dependson the distance ladder of cosmic which points to a tension 4.4σ [5] and the radiation data ofCMB (Cosmic Microwave Background) by Planck [2] are all at the level of background evolu-tion of the Cosmos. In opposite SDSS (the Sloan Digital Sky Survey) [6] and BAO (BaryonAcoustic Oscillations) analysis from BOSS (the Baryon Oscillation Spectroscopic Survey) [7]presents inconsistency at 2.5σ in H0 by Planck [8]. A discordance on σ8,0 of around 2.3σcould spot among Planck data [9, 10] and KiDS (Kilo-Degree Survey) [11] for perturbationsof matter. The exploration for new physics beyond the standard model and the source ofreflections on the gravity of ΛCDM , the ideas gathered from these surveys.

The quintessence [12], a common scalar field with appropriate potential minimally linkedto gravity and leads to inflation of late time is the next uncomplicated model recommendedfor DE. For the scalar field of spatially homogeneous quintessence, ωΛ > −1 is satisfied bythe EoS and thus can present accelerated expansion. It avoids the difficulties with the initialconditions [13] and also harmonious with its homogeneity, so this field is considered to beremarkably light. To illustrate the nature of DE except for quintessence many models of thescalar field have been introduced such as a scalar field with -ve kinetic energy, that presenta solution named as phantom DE [14], a scalar field model with the non-standard kineticterm [15, 16] known as k-essence, dilation [17] and tachyon [18]. The modified theories ofgravity known as f(R) gravity, where DE arises by the change of the geometrical part [19,20],interacting DE models [21, 22], DE models involving non-standard EoS [23, 24] and brane-world models [25,26]. Other recommended models which include DE (see [13] to review of allmentioned and other DE). Although the potential is chosen as support by phenomenologicalthoughtfulness in all mentioned scalar field models, which require the theoretical introduc-tion. Some data linked to the theory of quantum gravity, a framework of a fundamentaltheory is another solution of the DE problem identified as a principle of holographic [27–29]and generalized entropy formalism [30–32].

Recently, four new holographic dark energy models (HDE) [33–36] have been introducedto associate multiple generalized entropies of the FRW Universe horizon. In the presentwork, the focus is on the Tsallis holographic dark energy [33] inspired by the Tsallis en-tropy [30]. The point that practicing system horizon in the Tsallis statistics, we can achievethe Bekenstein entropy, which is the determination of all these efforts [37, 38]. The authorsalso exposed adequate stability by the concerned models [33–36]. To examine the cosmo-logical and gravitational systems of the generalized statistical mechanics [39–46] framework,positively a crucial role is performed by Tsallis definition of entropy [30]. Generally, thestudy of quantum gravity [47] confirms the conclusion that the content of the Tsallis entropysystem is a power-law function of the systems field [30].

Reminding the derivation and definition of the energy density ρΛ = 3c2m2p/L

2 of standardholographic depends on the connection S ∼ A ∼ L2 of entropy area of black holes, hereA = 4πL2 presents the horizon’s area [28]. Due to the consideration of quantum [48, 49], weget the modified HDE definition. In [30], Tsallis and Citro have exposed that we can modify

2

the black hole’s entropy of horizon as: Sδ = γAδ, where δ indicates the non-additivityparameter [30] and γ is an unknown constant. For the suitable limit of γ = 1/4G and δ = 1(in the link here c = h = kB = 1), the Bekenstein entropy is obtained. Particularly, thecommon probability of probability allocation [30] described, the system at this point or thepower-law of a probability distribution must have shifted to worthless. The quantum gravityhas confirmed the association [47] and have shown the impressive results in the holographicand cosmological setups [44, 45]. The holographic principle states that the scaling of thefreedom’s physical system of the number of degrees should exhibit through its bounding arearather than its volume [27,29], and the infrared cutoff should compel it, a connection amongUV (Λ) and the system entropy (S) and the IR (L) cutoffs is proposed by cohen et al. [28] as

L3Λ3 ≤ (S)3/4, (1)

after combining with Sδ = γAδ, it heads to [28], Λ4 ≤ (γ(4π)δ)L2δ−4, here Λ4 is the energydensity of the vacuum, in the hypothesis of HDE, it is DE’s energy density (ρΛ) [30, 50].Practicing this inequality the THDE density can be written as

ρΛ = BL2δ−4, (2)

here B is an unknown parameter [30, 36]. Considering the Hubble horizon as a flat FRWUniverse and for IR cutoff, a suitable applicant, besides no interaction among the otherelements of cosmos and the candidate of DE. Hence, the law of conservation and the energydensity matches to THDE in this way (L = H−1),

ρΛ = BH−2δ+4. (3)

For the HDE model [51], the authors presented a new infrared cut off named GO hori-zon cutoff, they have also represented the correspondence among the quintessence, dilation,tachyon, and k-essence energy density in the flat FRW Universe for HDE density. Towardaccelerated expansion, the correspondence also allows the dynamics and the potentials to re-construct the model of the scalar field. The researcher in [52] examined the conformity amongthe DBI-essence, tachyon, and the quintessence scalar field models in the core of chameleonBrans-Dicke cosmology with the model of NHDE. They have rebuilt the dynamics and thepotentials for the model of the scalar field in the connection of chameleon Brans-Dicke cosmol-ogy by concluding the appearance of Hubble parameter and energy density. They remarkedthat with the evolution of the Universe, the potential increases as the matter-chameleoncoupling becomes stronger. The conformity among the Chaplygin gas model, k-essence, thequintessence, dilation, and tachyon scalar field is shown in [53], in the non-flat Brans-DickeUniverse with the new HDE model. For this model, they reconstructed the dynamics andthe potentials. In the Chaplygin gas and new holographic quintessence model for acceleratedexpansion in Brans-Dicke cosmology, they have met some limitations to the related parame-ter of the model to the potentials. They have shown new results in the frame of Brans-Dickecosmology and also introduced special cases of new HDE in Einstein gravity. The researcherin [54] examined the generalization of the conformity between tachyon, dilation, k-essence,and the quintessence with the new HDE model in the Universe of non-flat FRW. They il-lustrated the accelerated expansion by reconstructing the dynamics and the potentials ofthese scalar field models. In the frame of Brans-Dicke cosmology, [55] inquired about the

3

cosmological relationship of interacting holographic energy density. They achieved the decel-eration and the EoS parameter of HDE in the Universe of non-flat. They combined HDE andBrans-Dicke field to support ωΛ = −1 loop for the EoS of non-interacting HDE. They formedEinstein field equations to transform ωΛ to the regime of the phantom when the interactionamong dark matter and dark energy was carried. In [56], for the Universe of non-flat FRWin the interaction of the new age graphic DE model, the correspondence among dilation,k-essence, and tachyon is studied. For the expansion of acceleration in the Universe, theyreplaced the dynamics and the potentials for the scalar field model. [57] investigated the cor-respondence in the Einstein gravity framework among k-essence, dilation, and tachyon scalarfield models with the interaction of the viscous ghost DE model. They reflected the Universeof non-flat FRW with interacting viscous ghost dark matter and dark energy. Accordingto the evolutionary performance of the interacting viscous ghost DE model, for scalar fieldmodels, they reformed the potentials and the dynamics that characterized the Universe’saccelerated expansion.

It is exciting to consider how the model of the scalar field can characterize the HDE asuseful theories if we see the holographic vacuum energy situation as tending towards theunderlying DE theory and observing an adequate explanation of the underlying DE theoryfor the models of the scalar field. [58, 59] reviewed the Chaplygin gas and the holographicphantom quintessence, [60, 61] presented the holographic tachyon models. All these modelsused the infrared cut off for future event horizon to reformed the conformity of the potentials.Though [62] shows the conflict in cut-offs. In [63], they achieved other techniques of recon-struction in theories with multiple or single scalar fields. Here in this work, I am interestedto represent the Tsallis holographic scenario of quintessence with an infrared cut off shownin Eq. (5) for the scalar field models as proposed in [64, 65] for the holographic DE. As weretaliate the constants δ and B, we can reconstruct the potentials for the quintessence fieldfor the THDE in f(R, T ) gravity.

The rest of the paper is organized as: we briefly review the f(R, T ) gravity in Sec. 2.The cosmological model and the THDE EoS parameter are discussed in Sec. 3. In Sec. 4,we construct the correspondence between the THDE model and the quintessence field. InSec. 5, we explain the correspondence with the Swampland criteria of the THDE model. Atlast in Sect. 6, we finish up our discussion.

2 Review on Gravitational field equations of f(R, T )

gravity

Prior to showing the present contribution taking into account the work referenced, let ushave a concise outline of the F (R, T ) gravity as the present work is going to investigate acosmological reconstruction in the structure of the F (R, T ) gravity. The theory proposes arevised gravity work performed through the action given in [66]

S =

∫

Lm

√−gd4x+1

16π

∫

f(R, T )√−gd4x, (4)

4

where, Lm is Lagrangian matter density. Also, R represents the ricci scalar and T rep-resents the trace of the matter energy momentum tensor (Tµν) for the arbitrary functionf(R, T ) given in Eq. (4). The matter’s stress-energy tensor is elaborated as [67]:

Tµν = − 2√−g

δ(√−gLm)

δgµν, (5)

Three exact stipulations of the functional anatomy of f(R, T ) have been considered in [66]as

f(R, T ) = 2f(T ) +R, f(R, T ) = f2(T ) + f1(R), f(R, T ) = f2(R)f3(T ) + f1(R) .

One can obtain various theoretical models for each choice of f because of f(R, T ) grav-ity’s field equations also depend on the matter’s field physical nature.

In the present contribution, we reflect the cosmological outcomes of the first class forwhich f(R, T ) = 2f(T ) +R. The field equation for this case by varying the action with gµνgiven by Eq. (4), is obtained as

Rµν −1

2Rgµν = 8πTµν + 2f

′

T (T )Tµν + f(T )gµν , (6)

here the prime expresses a derivative concerning the argument. The authors proposed thesefield equations to resolve the cosmological constant enigma [68]. By taking f(T ) as thefunction so that λT = f(T ), where constant is λ and allowing a dust Universe (ρ = T ,p = 0), the simplest model of cosmology can be achieved.

3 The cosmological model

By appropriating that the metric of the Universe is provided by the flat FRW metric,

ds2 = (dx2 + dy2 + dz2)a2(t)− dt2. (7)

The field equations of f(R, T ) gravity for the metric given by Eq. (7) are defined as

3a2

a2= ρΛ(3λ+ 8π), (8)

2a

a+

a2

a2= λρΛ. (9)

Hence, this model of f(R, T ) theory is comparable to a model of cosmology with aneffective cosmological constant Λeff ∝ H2, where H = a

ais the Hubble parameter [68].

Interestingly for the choice of f(R, T ), the form Geff = G± 2f′

(T ) of gravitational couplingbecomes time-dependent and an effective coupling. Displacing G through working on theparameter of the gravitational coupling, the interaction of gravitation among curvature andmatter is modified by the term 2f(T ) in the action of gravitation. Hence, the field equationreduces to a single equation of H [66], as

5

2H + 38π + 2λ

8π + 3λH2 = 0, (10)

general solution can be made as

H(t) =2(2λ+ 8π)

3(3λ+ 8π)

1

t. (11)

The scale factor unfolds agreeing to tβ = a(t), where β = 2(8π+2λ)3(8π+3λ)

and provides the power

law expansion a ∝ tβ. Since, we obtain the solution for the case of dark energy dominatedUniverse, the scale factor gives a matter dominated conduct for β = 2/3, showing that thederived model avoids conflict with the coincidence problem. The similar expression for theHubble parameter H was found in [51], while proposing the GO cutoff, for the small t limitin [69], and using the future event horizon cut off in [62]. From the equation of conservation,on the other side,

∂ρΛ∂t

+ (ρΛ + pΛ) 3H = 0, (12)

For the holographic energy practicing pressure densities pΛ = ωΛρΛ and EoS (equation ofstate) of a barotropic, we found an illustration for the EoS parameter ωΛ as

ωΛ = −1 +(2δ − 4)H

3H2. (13)

Using Eq. (11) in Eq. (13), we can find

ωΛ = −1− (δ − 2)

[

8π + 2λ

8π + 3λ

]

, (14)

Here, by use of ωΛ > −1, which is the allowed range for quintessence [13], we can obtain theconstraints of the parameters δ and λ. If ωΛ > −1, the rate of expansion of the Universeaccelerates always [13]. In Eq. (14), the EoS ωΛ is explained in terms of the constants λ andδ. The constants λ and δ must justify the restrictions followed from Eq. (14), to get accel-erated expansion. If −6π < λ < −4π and 0 < δ < 2 then we consider the ωΛ > −1 phaseand for −6π < λ < −4π and δ > 2 the density of Tsallis holographic shows the evolution’sphase as a phantom with ωΛ < −1. While, for λ = −4π or δ = 2, the EoS ωΛ = −1, mimicsthe cosmological constant.

It is necessary to specify that differentiating the frame of Tsallis HDE with the standardHDE, Tsallis HDE provides more extreme behavior and quantifies at the presence of thenovel parameter δ. One can obtain the sub-case of the standard HDE framework because ofits agreeable foundation, which is distinct for δ = 1 [70]. One should perform many moreresearches for the description of nature, before the circumstances considered as a profitablecontender. Importantly, for compelling the model parameters, one should strike out sharedempiric research using large scale structure and data from CMB at each confusion backgroundand levels. To classify the Universal highlights of the framework in modern times, we oughtto perform the analysis of phase space in detail, the underlying conditions independently,

6

Figure 1: The behaviour of Tsallis holographic quintessence scalar field (φ) against redshift(z) for THDE 1 and λ = −5π.

and the specific evolving [70].

Approaching, Universe’s accelerated expansion form one of the current works [71], theresearcher recognized the legitimacy of the holographic principle. Based on the entropy ofTsallis, the researcher explained the highlights of the DE model, which says of the BG en-tropy’s non-additive generalization that needed to use in the scientific summary of some nonextensive frames, such as gravitation. The researcher reflected on the IR cutoff for the futureevent horizon in this research. This choice made by the researcher has allowed him to includestandard thermodynamics and standard HDE as sub-classes and to avoid the cosmologicalvariations. The researcher has divided his work into two distinct THDE conditions: theprimary model recovers standard HDE with a fixed value of B = 3 in the limit of δ = 1,while they left B as an independent parameter for the secondary model. Subsequently, theytried to make a correlation among the latest observation of cosmology and the observationof earlier specified hypothetical models [71].

In [71], the researcher had practiced the data from the GRF (growth rate factor) matterfluctuations and combined it with the data from type SN (Ia Supernovae) and OHD (observa-tional Hubble data) to restrain Tsallis DE model as IR cutoff for the future event horizon andobtained 68% (95%) confidence level and possibilities of the parameter of cosmology assum-ing from the study of MCMC (Markov Chain Monte Carlo) with independent B and fixedB = 3 of THDE models. δ the nonextensive Tsallis parameter, and B the model parameterare constraints according to the research of the observational data for Tsallis DE model asδ = 0.939

+0.053(0.107)−0.054(0.101) and B = 3, and B = 2.864

+0.741(2.405)−1.454(1.821) and δ = 0.941

+0.053(0.104)−0.054(0.101) [71].

In [72], they have examined briefly about the observational limitations of THDE modelas IR cutoff with Hubble horizon. They applied various observational data of cosmology,BAO ( Baryon acoustic oscillation), data of the Gamma-Ray Burst, the Pantheon SN (type

7

Ia supernovae), the confined value of H0 (Hubble constant) and CMB (cosmic microwavebackground) for restraining independent parameter. B = 3 and δ = 1.871+0.190

−0.441 are mostcommendable results in [72].

As restrained in [71,72], we used the model parameter’s value and expressed it as THDE1 and THDE 2 in my research work.

THDE 1

For the current contribution, we present the model parameter’s value as δ = 1.3, 1.5, 1.7,1.9 and B = 3 in the primary framework.

THDE 2

For the current contribution, we present the model parameter’s value as δ = 1.3, 1.5, 1.7,1.9 and B = 2.8 in the secondary framework.

It is also proposed in [70], that there are extensive limits for δ and B which can produceaspired results. Comparing the results with those of [70–72], we can easily see that the valuesof δ and B used in this work place within the allowed interval of δ and B reported by [70–72].

Figure 2: The behaviour of Tsallis holographic DE quintessence potential against redshift(z) for THDE 1 and λ = −5π.

4 Reconstruction of quintessence model for THDE

In this segment, we shall investigate the correspondence between the quintessence and theTHDE in a flat f(R, T ) Universe for the case THDE 1 and THDE 2. To establish the cor-

8

respondence between the quintessence and the THDE model, we compare the equation ofstate for the quintessence field with the THDE model given by Eq. (14), and also equatethe THDE model energy density given by Eq. (3) with the corresponding energy densityof quintessence scalar field model. Furthermore, we shall reconstruct the dynamics and thepotentials for the quintessence model. We can produce the corresponding outcomes of the po-tentials and the quintessence scalar field for the both cases of THDE model in f(R, T ) gravity.

THDE 1

The pressure and the energy density of the quintessence field in FRW background [13]

ρφ = V (φ) +1

2φ2, (15)

pφ = −V (φ) +1

2φ2 (16)

For the scalar field, the equation of state (EoS) is given as

ωφ =−2V (φ) + φ2

2V (φ) + φ2, (17)

which on comparing with the Tsallis holographic EoS parameter Eq. (14), provides theequation

−2V (φ) + φ2

2V (φ) + φ2= −1 − (δ − 2)

[

8π + 2λ

8π + 3λ

]

(18)

which together with the equation

ρφ = V (φ) +1

2φ2 = BH−2δ+4 (19)

to lead the exact expressions for the potential and the scalar field, this equation can besolved, specifically,

φ = ±√

B(2− δ)

δ − 1

(

2

3

)

−δ+2[8π + 3λ

8π + 2λ

]−2δ+3

2

tδ−1. (20)

The expression given by Eq. (20) represents the scalar fied and is plotted in Fig. 1. Here,we have taken the +ve sign in Eq. (20). We observe from this figure that with the increasein the red shift z, the scalar field φ decreases, displays constant at the high red shift and φbecomes finite at future. The similar behavior was observed for the scalar field φ in [73] forthe k-essence scalar field [74]. Using Eq. (11) in Eq. (19) and then solving with Eq. (18),we can get the expression for the potential in terms of scalar field φ as

V (φ) = B

(

2

3

)

−2δ+4 [

λ+ 4δπ + δλ

8π + 3λ

] [

8π + 3λ

8π + 2λ

]

−2δ+4

A (21)

A =

[

√

B(2− δ)

φ(δ − 1)

(

2

3

)

−δ+2 [

8π + 3λ

8π + 2λ

]−2δ+3

2

]−2δ+4

δ−1

9

Figure 3: The behavior of Tsallis holographic DE quintessence potential against the scalarfield (φ) for THDE 1 and λ = −5π.

We have plotted the quintessence potential with red shift z in Fig. 2, for the THDE 1model. We observe from Fig. 2 that the potential decreases as we proceed from past tofuture. The similar behavior for the quintessence potential has been observed in [64] for theholographic quintessence model, and for the holographic tachyon model in [60]. We have alsoplotted the quintessence potential versus the scalar field as exhibited in Fig. 3.

It is clearly seen from Fig. 3, that the potential decreases as the scalar field increases forthe THDE 1 model. The same behavior was shown in [75] for the quintessence potentials. Ifwe take the negative sign in Eq. (20), we can too evaluate the potential and at present. InFig. 4, we can see the shift of the scalar field, because of this result, it can not modify theshape of the potential, and also can not influence the cosmological evolution of the Universe.In this case, we can see the appearance of the quintessence field in Fig. 5, by replacing thesign of the scalar field. One can see from Fig. 5, the potential is more steep in the past,approaching to be flat at future implies that the potential is rolled down by the quintessencefield as the Universe expand. The same conclusion was observed in [64] for the quintessencepotential. Therefore, it was proposed in [75] that the potentials are of a runway type, anddecrease as the Universe expands.

THDE 2

Copying the same style as B = 3, we will receive the shape of the potential and the scalarfield that recognizes the accelerated expansion for the case B = 2.8. In this case, Fig. 6imitates the scalar field in terms of red shift, Fig. 7 draws the potential in terms of red shift,and Fig. 8 portrays the potential in terms of the scalar field for the quintessence field.

As in Eq. (20), we turn the negative sign of the scalar field, we can see the shift of thescalar field in Fig. 9, and the shift of the potential versus the scalar field in Fig. 10 for thequintessence field. It is remarked that the dynamics of the quintessence behave in the same

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Figure 4: The behaviour of Tsallis holographic DE quintessence scalar field against redshift(z) for THDE 1 and λ = −5π. Here, the negative sign is taken in Eq. (20).

Figure 5: The behavior of the Tsallis holographic DE quintessence potential against thescalar field (φ) for the THDE 1 and λ = −5π. Here, the negative sign is taken in Eq. (20).

11

Figure 6: The behaviour of the Tsallis holographic DE quintessence scalar field againstredshift (z) for the THDE 2 and λ = −5π.

Figure 7: The behavior of Tsallis holographic DE quintessence potential against red shift (z)for THDE 2 and λ = −5π.

12

Figure 8: The behaviour of the Tsallis holographic DE quintessence potential against thescalar field (φ) for the THDE 2 and λ = −5π.

manner as in the B = 3 case.

5 The correspondence with the Swampland criteria of

quintessence phase for THDE model

In modern times a new interest is developed to connect applicable models of cosmology withgravity’s quantum theory, which can easily differentiate between the effective field models re-lated to the nominal string landscape. Recently Swampland criteria are introduced to choosefield potentials effectively. Precisely, the excitement aroused because accurate solutions of deSitter with a positive constant of cosmology is not fit with the string landscape, which meansit cannot link at the arrangement of high energy with some fundamental theory. Lately, theresearcher investigated the connection of the Swampland conjecture and the viability of thef(R) theory [76]. Several philosophical contexts reviewed, the DE phase as we have explainedin the introduction of the paper, for example, quintessence models, modified gravity, and soon, and the hurdle to get the compatibility with the observational data represented.

Newly, [77, 78] derived the swampland criteria of the string theory constrained with thequintessence model of DE. Approaching the Swampland criteria in the connection of DE’squintessence models, many papers reported after the work in [77,78], for example, [76,79–89].The Swampland criteria restrain and rules on those field theories which are conflicting withquantum gravity [90]. Various researchers submitted the work satisfying the low energyeffective theory [91] for the scalar field in the Swampland conjecture,

Conjecure 1 : | ∆φ |≤ MPd1, (22)

13

Figure 9: The behavior of the Tsallis holographic DE quintessence scalar field against thered shift (z) for the THDE 2 and λ = −5π. Here, the negative sign is taken in Eq. (20).

Figure 10: The behavior of the Tsallis holographic DE quintessence potential against thescalar field (φ) for the THDE 2 and λ = −5π. Here, the negative sign is taken in Eq. (20).

14

Conjecure 2 : | ∆V |≥ d2V

MP. (23)

If the conjecture has UV (ultraviolet) completion compatible with quantum gravity, thenover a specific limit of the scalar field the +ve constants d2 and d1 are of Ist order, and theconjecture lies in the Swampland criteria, and also the potential of scalar is too flat.

In the analysis of DE, the quintessence models plays an important role, therefore in thiswork, we have examined the performance of the THDE model in the connection of Swamplandconjecture for quintessence case. Eq. (20) shows the scalar field φ that is associated withswampland criteria 1, and Eq. (21) shows the scalar potential V that is associated withswampland criteria 2. The scalar field φ in the terms of red shift (z) and the potential fieldV in the terms of the scalar field φ are framed for both the models THDE 1 and THDE 2shown in Fig. 1-10. It is shown in these Figs. that the potential field is +ive while the scalarfield is not at minimum, as shown in [79] for the models of quintessence.

6 Discussion

In this work, we have brought up a comprehensive analysis for the cosmological evolution ofthe Tsallis holographic quintessence model of DE in the framework f(R, T ) gravity with theinfrared cut off as Hubble horizon. In this model, we have three parameters δ, λ, and B.The parameter δ performs a major role in the Tsallis holographic evolution of the Universe.We have investigated ωΛ > −1 phase, if 0 < δ < 2 and −6π < λ < −4π and the Tsallisholographic EoS represents the phase of evolution as a phantom with ωΛ < −1 for δ > 2 and−6π < λ < −4π. While, the EoS of the THDE copies the cosmological constant ωΛ = −1for δ = 2 or λ = −4π. We have reconstructed the potential and the scalar field using theconformity with the Tsallis holographic density and with the equation of state parameter inthe state of quintessence field for both the cases of the THDE model i.e. the THDE 1 and theTHDE 2 in the region ωΛ > −1 which is the defined region for the quintessence era. FromFigs. 1-5, we may observe the quintessence field dynamics for the THDE 1 model. Fig. 5,depicts that at the early times the potential is steep and turning flat at the present and at fu-ture. Hence, based on the Hubble horizon cut off, the kinetic term is continuously decreasingfor the Tsallis holographic quintessence model as witnessed in [64,92] for the negative sign inthe expression of Eq. (20). For the positive sign in the expression of Eq. (20), Fig. 3 showsthe appearance of the potential and relates this work to [75]. We have achieved the sameresult for the reconstruction of Tsallis quintessence potential as achieved in [93] and [94]. Inboth instances, Fig. 3 and Fig. 5 with the expansion in the Universe, the potential decreasesas presented in Fig. 2. The reconstruction for the case of the THDE 2, Figs. 6-10 gives thecomparable result as the case THDE 1. We additionally examine the swampland conjecturefor the Tsallis HDE model and the outcome is fascinating. Particularly observed that thepotential (V (φ)) is asymptotically flat as Universe expands.

According to the results presented in the literature, the reconstruction is beneficial inrepresenting the main characteristics of the potential for the quintessence model. Althoughthe Tsallis holographic model, which depends on the parameters, lifts the reconstruction un-ambiguously and immediately, which is shown phenomenological, still one should understand

15

and review the hypothetical root of Tsallis holographic density. Nevertheless, we appropri-ately understand through the obtained result about the interesting problems of the cosmology.

Based on the model, to explain Tsallis holographic dark energy the reconstruction ofthe phantom, tachyon, k-essence and dilation scalar field can show detailed or meaningfuleffects on the evolution of the Universe. In the future paper, we shall spout these studies.As witnessing the Universe’s future, my work in this paper contributes a different and new,reasonably pleasing opportunity to understand the nature of the THDE in the modifiedgravity.

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